Thorsten M. BuzugComputed Tomography

Thorsten M. Buzug

Computed Tomography

From Photon Statisticsto Modern Cone-Beam CT

With 475 Figures and 10 Tables

123

ThorstenM. Buzug, Prof. Dr.Institut für MedizintechnikUniversität zu LübeckRatzeburger Allee LübeckGermanyE-mail: buzug@imt.uni-luebeck.de

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Preface

This book provides an overview of X-ray technology, the historic developmentalmilestones of modern CT systems, and gives a comprehensive insight into the mainreconstruction methods used in computed tomography. The basis of reconstruc-tion is, undoubtedly, mathematics. However, the beauty of computed tomographycannot be understood without a detailed knowledge of X-ray generation, photon–matter interaction, X-ray detection, photon statistics, as well as fundamental signalprocessing concepts and dedicated measurement systems.Therefore, the reader willfind a number of references to these basic disciplines together with a brief introduc-tion to the underlying principles of CT.

This book is structured to cover the basics of CT: from photon statistics tomod-ern cone-beam systems. However, the main focus of the book is concerned with de-tailed derivations of reconstruction algorithms in D and modern D cone-beamsystems. A thorough analysis of CT artifacts and a discussion of practical issues,such as dose considerations, provide further insight intomodernCT systems.Whilemainly written for graduate students in biomedical engineering, medical engineer-ing science, medical physics, medicine (radiology), mathematics, electrical engin-eering, and physics, experienced practitioners in these fields will benefit from thisbook as well.

The didactic approach is based on consistent notation. For example, the nota-tion of computed tomography is used in the signal processing chapter. Therefore,contrary to many other signal processing books, which use time-dependent values,this book uses spatial variables in one, two or three dimensions. This facilitatesthe application of the mathematics and physics learned from the earlier chaptersto detector array signal processing, which is described in the later chapters. Add-itionally, special attention has been paid to creating a text with detailed and richlydiscussed algorithm derivations rather than compact mathematical presentations.The concepts should give even undergraduate students the chance to understandthe principal reconstruction theory of computed tomography.The text is supportedby a large number of illustrations representing the geometry of the projection sit-uation. Since the impact of cone-beam CT will undeniably increase in the future,three-dimensional reconstruction algorithms are illustrated and derived in detail.

This book attempts to close a gap. There are several excellent books on medicalimaging technology that give a comprehensive overview of modern X-ray technol-ogy, computed tomography, magnetic resonance imaging, ultrasound, or nuclear

VI Preface

medicine modalities like PET and SPECT. However, these books often do not gointo the mathematical detail of signal processing theory. On the other hand, thereare a number of in-depth mathematical books on computed tomography that donot discuss practical issues. The present book is based on the German book Ein-führung in die Computertomographie, which first appeared during the summer of. Fortunately, since the book was used by many of my students in lectures onEngineering in Radiology,Medical Engineering, Signals and Systems inMedicine, andTomographic Methods, I received a lot of feedback regarding improvements on thefirst edition. Therefore, the idea arose to publish an English version of the book,which is a corrected and extended follow-up.

I would like to thank Siemens Medical Solutions, General Electric Medical Sys-tems, and Philips Medical Systems, who generously supported my laboratories inthe field of computed tomography. In particular, I would like to thank my friendDr. Michael Kuhn, former Director of Philips Research Hamburg. It was his ini-tiative that made possible the first installation of CT in my labs in . Addition-ally, I have to thank Mrs. Annette Halstrick and Dr. Hans-Dieter Nagel (PhilipsMedical Systems Hamburg), Leon de Vries (Philips Medical Systems Best), DorisPischitz, Jürgen Greim and Robby Rokkitta (Siemens Medical Solutions Erlangen),Dieter Keidel and Jan Liedtke (General Electric Medical Systems) for many photosin this book. I would like to thankWolfgangHärer (AXICC, SiemensMedical Solu-tions), Dr. Gerhard Brunst, (General Electric Medical Systems), Dr. ArminH. Pfoh,Director of General Electric Research Munich, Dr. Wolfgang Niederlag (HospitalDresden-Friedrichstadt), Prof. Dr. Heinz U. Lemke (Technical University Berlin),Dr. Henrik Turbell (Institute of Technology, Linköpings Universitet), and Prof.Dr.Dr. Jürgen Ruhlmann (Medical Center Bonn) for the courtesy to allowme to usetheir illustrations and photos. Further, I have to thank the Digital Collections andArchives of Tufts University, the Collection of Portraits of the Austrian Academy ofSciences, and the Röntgen-Kuratorium Würzburg e.V. for the courtesy to allow meto use their photos.

Additionally, I have to thank my friends, colleagues, and students for proof-reading and translating parts of the book. In alphabetical order I appreciated thehelp of:

Dr. Bernd David (Philips Research Laboratories Hamburg)Katie Dechambre (Milwaukee School of Engineering)Erin Fredericks (California Polytechnic State University, San Luis Obispo)Sebastian Gollmer (University of Lübeck)Dr. Franko Greiner (University of Kiel)Tobias Knopp (University of Lübeck)Dieter Lukhaup (Schriesheim-Altenbach)Andreas Mang (University of Lübeck)Prof. Dr.-Ing. Alfred Mertins (University of Lübeck)Jan Müller (University of Lübeck)Dr. Hans-Dieter Nagel (Philips Medical Systems Hamburg)Susanne Preissler (RheinAhrCampus Remagen)

Preface VII

Tony Shepherd (University College London)Vyara Tonkova (RheinAhrCampus Remagen)

Many special thanks go to Sebastian Gollmer, Andreas Mang, and Jan Müller, whodid the copy editing of the complete manuscript. However, for the errors that re-main, I alone am responsible and apologize in advance.

I would like to thank the production team at le-tex as well as Paula Francis forcopy editing. Further, I have to thank Springer Publishing, especially Dr. Ute Heil-mann and Wilma McHugh for their excellent cooperation over the last few years.

Finally, I would like to thank my wife Kerstin, who has supported and sustainedmy writing efforts over the last few years. Without her help, patience, and encour-agement this book would not have been completed.

Lübeck, June Thorsten M. Buzug

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Computed Tomography – State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . 1. Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Structure of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Fundamentals of X-ray Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. X-ray Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

.. X-ray Cathode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.. Electron–Matter Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.. Temperature Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.. X-ray Focus and Beam Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.. Beam Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.. Special Tube Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

. Photon–Matter Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.. Lambert–Beer’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.. Mechanisms of Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

. Problems with Lambert–Beer’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46. X-ray Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

.. Gas Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.. Solid-State Scintillator Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . 50.. Solid-State Flat-Panel Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

. X-ray Photon Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59.. Statistical Properties of the X-ray Source . . . . . . . . . . . . . . . . . . . 60.. Statistical Properties of the X-ray Detector . . . . . . . . . . . . . . . . . 64.. Statistical Law of Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66.. Moments of the Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . 68.. Distribution for a High Number of X-ray Quanta . . . . . . . . . . . 70.. Non-Poisson Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

X Contents

3 Milestones of Computed Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75. Tomosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76. Rotation–Translation of a Pencil Beam (First Generation) . . . . . . . . . . . 79. Rotation–Translation of a Narrow Fan Beam (Second Generation) . . . 83. Rotation of a Wide Aperture Fan Beam (Third Generation) . . . . . . . . . 84. Rotation–Fix with Closed Detector Ring (Fourth Generation) . . . . . . . 87. Electron Beam Computerized Tomography . . . . . . . . . . . . . . . . . . . . . . . 89. Rotation in Spiral Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90. Rotation in Cone-Beam Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91. Micro-CT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93. PET-CT Combined Scanners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96. Optical Reconstruction Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4 Fundamentals of Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102. Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102. Fundamental Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102. Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

.. Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.. Position or Translation Invariance . . . . . . . . . . . . . . . . . . . . . . . . . 105.. Isotropy and Rotation Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . 105.. Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.. Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

. Signal Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106. Dirac’s Delta Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109. Dirac Comb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112. Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115. Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116. Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118. Convolution Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124. Rayleigh’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125. Power Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125. Filtering in the Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126. Hankel Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128. Abel Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132. Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133. Sampling Theorem and Nyquist Criterion . . . . . . . . . . . . . . . . . . . . . . . . . 135. Wiener–KhintchineTheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141. Fourier Transform of Discrete Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144. Finite Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145. z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147. Chirp z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Contents XI

5 Two-Dimensional Fourier-Based Reconstruction Methods . . . . . . . . . . . . . . . 151. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151. Radon Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153. Inverse Radon Transformation and Fourier Slice Theorem . . . . . . . . . . 163. Implementation of the Direct Inverse Radon Transform . . . . . . . . . . . . 167. Linogram Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170. Simple Backprojection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175. Filtered Backprojection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179. Comparison Between Backprojection and Filtered Backprojection . . . 183. Filtered Layergram: Deconvolution of the Simple Backprojection . . . . 187. Filtered Backprojection and Radon’s Solution . . . . . . . . . . . . . . . . . . . . . . 191. Cormack Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6 Algebraic and Statistical Reconstruction Methods . . . . . . . . . . . . . . . . . . . . . . . 201. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201. Solution with Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . 207. Iterative Reconstruction with ART . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211. Pixel Basis Functions and Calculation of the SystemMatrix . . . . . . . . . 218

.. Discretization of the Image: Pixels and Blobs . . . . . . . . . . . . . . . 219.. Approximation of the SystemMatrix in the Case of Pixels . . . . 221.. Approximation of the SystemMatrix in the Case of Blobs . . . . 222

. Maximum Likelihood Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.. Maximum Likelihood Method for Emission Tomography . . . . 224.. Maximum Likelihood Method for Transmission CT . . . . . . . . . 230.. Regularization of the Inverse Problem . . . . . . . . . . . . . . . . . . . . . 235.. Approximation Through Weighted Least Squares . . . . . . . . . . . . 238

7 Technical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241. Reconstruction with Real Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

.. Frequency Domain Windowing . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.. Convolution in the Spatial Domain . . . . . . . . . . . . . . . . . . . . . . . . 247.. Discretization of the Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

. Practical Implementation of the Filtered Backprojection . . . . . . . . . . . . 255.. Filtering of the Projection Signal . . . . . . . . . . . . . . . . . . . . . . . . . . 255.. Implementation of the Backprojection . . . . . . . . . . . . . . . . . . . . . 258

. Minimum Number of Detector Elements . . . . . . . . . . . . . . . . . . . . . . . . . 258. Minimum Number of Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259. Geometry of the Fan-Beam System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261. Image Reconstruction for Fan-Beam Geometry . . . . . . . . . . . . . . . . . . . . 262

.. Rebinning of the Fan Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.. Complementary Rebinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

XII Contents

.. Filtered Backprojection for Curved Detector Arrays . . . . . . . . . 272.. Filtered Backprojection for Linear Detector Arrays . . . . . . . . . . 280.. Discretization of Backprojection for Fan-Beam Geometry . . . . 286

. Quarter-Detector Offset and Sampling Theorem . . . . . . . . . . . . . . . . . . . 293

8 Three-Dimensional Fourier-Based Reconstruction Methods . . . . . . . . . . . . . . 303. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303. Secondary Reconstruction Based on 2D Stacks of Tomographic Slices 304. Spiral CT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309. Exact 3D Reconstruction in Parallel-Beam Geometry . . . . . . . . . . . . . . 321

.. 3D Radon Transform and the Fourier Slice Theorem . . . . . . . . . 321.. Three-Dimensional Filtered Backprojection . . . . . . . . . . . . . . . . 326.. Filtered Backprojection and Radon’s Solution . . . . . . . . . . . . . . . 327.. Central Section Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.. Orlov’s Sufficiency Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

. Exact 3D Reconstruction in Cone-Beam Geometry . . . . . . . . . . . . . . . . 336.. Key Problem of Cone-Beam Geometry . . . . . . . . . . . . . . . . . . . . 339.. Method of Grangeat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.. Computation of the First Derivative on the Detector . . . . . . . . . 347.. Reconstruction with the Derivative of the Radon Transform . . 348.. Central Section Theorem and Grangeat’s Solution . . . . . . . . . . . 350.. Direct 3D Fourier Reconstruction with the Cone-Beam

Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.. Exact Reconstruction using Filtered Backprojection . . . . . . . . . 357

. Approximate 3D Reconstructions in Cone-Beam Geometry . . . . . . . . . 366.. Missing Data in the 3D Radon Space . . . . . . . . . . . . . . . . . . . . . . 366.. FDK Cone-Beam Reconstruction for Planar Detectors . . . . . . . 371.. FDK Cone-Beam Reconstruction for Cylindrical Detectors . . . 388.. Variations of the FDK Cone-Beam Reconstruction . . . . . . . . . . 390

. Helical Cone-Beam Reconstruction Methods . . . . . . . . . . . . . . . . . . . . . . 394

9 Image Quality and Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403. Modulation Transfer Function of the Imaging Process . . . . . . . . . . . . . . 404. Modulation Transfer Function and Point Spread Function . . . . . . . . . . 410. Modulation Transfer Function in Computed Tomography . . . . . . . . . . 412. SNR, DQE, and ROC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421. 2D Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

.. Partial Volume Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.. Beam-Hardening Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.. Motion Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.. Sampling Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.. Electronic Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.. Detector Afterglow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.. Metal Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

Contents XIII

.. Scattered Radiation Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443. 3D Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

.. Partial Volume Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.. Staircasing in Slice Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448.. Motion Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450.. Shearing in Slice Stacks Due to Gantry Tilt . . . . . . . . . . . . . . . . . 451.. Sampling Artifacts in Secondary Reconstruction . . . . . . . . . . . . 454.. Metal Artifacts in Slice Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.. Spiral CT Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.. Cone-Beam Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458.. Segmentation and Triangulation Inaccuracies . . . . . . . . . . . . . . . 459

. Noise in Reconstructed Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.. Variance of the Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . 462.. Variance of the Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.. Dose, Contrast, and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

10 Practical Aspects of Computed Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471. Scan Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471. Data Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

.. Hounsfield Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.. WindowWidth and Window Level . . . . . . . . . . . . . . . . . . . . . . . . 476.. Three-Dimensional Representation . . . . . . . . . . . . . . . . . . . . . . . 479

. Some Applications in Medicine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

11 Dose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485. Energy Dose, Equivalent Dose, and Effective Dose . . . . . . . . . . . . . . . . . 486. Definition of Specific CT Dose Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 487. Device-RelatedMeasures for Dose Reduction . . . . . . . . . . . . . . . . . . . . . 493. User-RelatedMeasures for Dose Reduction . . . . . . . . . . . . . . . . . . . . . . . 499

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

1 Introduction

Contents

1.1 Computed Tomography – State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Structure of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1Computed Tomography – State of the Art

Computed tomography (CT) has evolved into an indispensable imaging methodin clinical routine. It was the first method to non-invasively acquire images of theinside of the human body that were not biased by superposition of distinct anatom-ical structures. This is due to the projection of all the information into a two-dimensional imaging plane, as typically seen in planar X-ray fluoroscopy.Therefore,CT yields images of much higher contrast compared with conventional radiogra-phy. During the s, this was an enormous step toward the advance of diagnosticpossibilities in medicine.

However, research in the field of CT is still as exciting as at the beginning of itsdevelopment during the s and s; however, several competing methods ex-ist, the most important being magnetic resonance imaging (MRI). Since the inven-tion ofMRI during the s, the phasing out of CT has been anticipated. Neverthe-less, to date, the most widely used imaging technology in radiology departments isstill CT. AlthoughMRI and positron emission tomography (PET) have been widelyinstalled in radiology and in nuclear medicine departments, the term tomographyis clearly associated with X-ray computed tomography .

Some hospitals actually replace their conventional shock rooms with a CT-based virtual shock room. In this scenario, imaging and primary care of the patienttakes place using a CT scanner equipped with anesthesia devices. In a situation

In the United States computed tomography is also called CAT (computerized axial tom-ography).

1 Introduction

where the fast three-dimensional imaging of a trauma patient is necessary (and it isunclear whetherMRI is an adequate imaging method in terms of compatibility withthis patient), computed tomography is the standard imagingmodality. Additionally,due to its ease of use, clear interpretation in terms of physical attenuation values,progress in detector technology, reconstruction mathematics, and reduction of ra-diation exposure, computed tomography will maintain and expand its establishedposition in the field of radiology.

Furthermore, the preoperatively acquired CT image stack can be used to syn-thetically compute projections for any given angulations. A surgeon can use thisinformation in order to get an impression of the images that are taken intraoper-atively by a C-arm image intensifier. Therefore, there is no need to acquire add-itional radiographs and the artificially generated projection images actually resem-ble conventional radiographs. Additionally, the German Employer’s Liability Insur-ance Association insists on aCT examination in severe accidents that occur at work.Therefore, CT has advanced to become the standard diagnostic imagingmodality intrauma clinics. Patients with heavy trauma, fractures, and luxations benefit greatlyfrom the clarification provided by imaging techniques such as computed tomog-raphy.

Recently, interesting technical, anthropomorphic, forensic, and archeological(Thomsen et al. ) as well as paleontological (Pol et al. ) applications ofcomputed tomography have been developed.These applications further strengthenthe method as a generic diagnostic tool for non-destructive material testing andthree-dimensional visualization beyond its medical use. Magnetic resonance imag-ing fails whenever the object to be examined is dehydrated. In these circumstances,computed tomography is the three-dimensional imaging method of choice.

1.2Inverse Problems

The mathematics of CT image reconstruction has influenced other scientific fieldsand vice versa. The backprojection technique, for instance, is used in both geo-physics and radar applications (Nilsson ). Clearly, the fundamental problemof computed tomography can be easily described: Reconstruct an object from itsshadows or, more precisely, from its projections. An X-ray source with a fan- orcone-beam geometry penetrates the object to be examined as a patient in medicalapplications, a skull found in archeology or a specimen in nondestructive testing(NDT). In the so-called third generation scanners, the fan-shaped X-ray beam fullycovers a slice section of the object to be examined.

Depending on the particular paths, the X-rays are attenuated at varying extentswhen running through the object; the local absorption is measured with a detectorarray. Of course, the shadow that is cast in only one direction is not an adequatebasis for the determination of the spatial distribution of distinct structures insidea three-dimensional object. In order to determine this structure, it is necessary to

1.2 Inverse Problems

irradiate the object from all directions. Figure . schematically illustrates this prin-ciple, where pγi (ξ) represents the attenuation profile of the beam versus the X-raydetector array coordinate ξ under a particular projection angle γi . If the differentattenuation or absorption profiles are plotted over all angles of rotation γi of thesampling unit, a sinusoidal arrangement of the attenuation or projection integralvalues is obtained. In two dimensions, these data, pγi (ξ), represent the radon spaceof the object, which is essentially the set of raw data.

In a special CT acquisition protocol, the spiral measurement process, theX-ray tube is continuously rotated, with the examination table being moved lin-early through the measuring field. This scan process produces data that take a spi-ral or helical path. This method offers the possibility of computing any number ofslices retrospectively, so that an accurate three-dimensional rendering can be ex-pected. To allow this acquisition technique, a slip ring transfer system has been de-veloped. In such a system, power supply to the X-ray tube and signal transfer fromthe detector system is guaranteed, even though the imaging system continuouslyrotates.

From a mathematical point of view, image reconstruction in computed tomog-raphy is the task of computing the spatial structure of an object that casts shadowsusing these very shadows. The solution for this problem is complex and involvestechniques in physics, mathematics, and computer science. The described scenariois referred to as the inverse problem in mathematics.

Fig. .. Schematic illustration of computed tomography (CT). Three homogeneous objectswith quadratic intersection areas are exposed with X-ray under the projection angles γand γ. Each projection angle produces a specific shadow, pγ(ξ), which, measured withthe detector array represents the integral X-ray attenuation profile. The geometric shadowboundaries are indicated with dashed lines. However, analysis of the profile under the firstprojection angle, γ , on its own does not allow one to deduce an estimate of the number ofseparated objects

1 Introduction

1.3Historical Perspective

A particular kind of mathematical problem in CT became popular in the swhen the astrophysicist Bracewell proved that the resolution of telescopes couldbe significantly improved if spatially distributed telescopes are appropriately syn-chronized. However, in , similar problems with the same mathematical basishad already been discussed (cf. Cormack [] and the references therein; manyexamples are collected by Deans []).

One example can be found in the field of statistics: Given all marginal distri-butions of a probability distribution density, is it possible to deduce the probabilitydensity itself? In this example, the marginal distributions represent an equivalentexample for the measured projections in computed tomography. Another examplecan be found in astrophysics: Looking from earth into a particular direction of theuniverse, only the radial velocity component of the stars can be obtained via thespectral Doppler red shift. This again represents the same inverse problem that hasto be solved if the distribution of the actual three-dimensional velocity vector is tobe reconstructed from the radial velocity components acquired from all availabledirections.

In computed tomography, the meaning of the mathematical term inverse prob-lem is immediately apparent. In contrast to the situation shown in Fig. ., the spa-tial distribution of the attenuating objects that produce the projection shadow isnot known a priori. This, actually, is the reason for acquiring the projections alongthe rotating detector coordinate ξ over a projection angle interval of at least �.Figure . illustrates this situation. It is an inversion of integral transforms. Froma sequence of measured projection shadows �pγ(ξ), pγ(ξ), pγ(ξ), . . .�, the spa-

Fig. .. Schematic illustration of the inverse problem posed by CT. Attenuation pro-files, pγ(ξ), have been measured for a set of projection angulations, γ and γ.The unknowngeometry, or the object with its associated spatial distribution of attenuation coefficients, hasto be calculated from a complete set of attenuation profiles �pγ(ξ), pγ(ξ), pγ(ξ), . . .�

1.3 Historical Perspective

Fig. ..Left: AllanMacLeodCormack (–) shortly after the official announcement ofthe Nobel Prizes for medicine in (courtesy of Tufts University, Digital Collections andArchives). Right: Sir Godfrey Hounsfield (–) in front of his first EMI CT scanner(courtesy of General Electric Medical Systems)

Fig. .. Left: Johann Radon (–; courtesy of the Austrian Academy of Sciences[OAW], collection of portraits). Right: Wilhelm Conrad Röntgen (–; courtesy ofRöntgen-KuratoriumWürzburg e.V.)

1 Introduction

tial distribution of the objects, or more precisely, the spatial distribution of the at-tenuation coefficients within a chosen section through the patient, must be esti-mated.

In , the solution to this problem was applied for the first time to a sequenceof X-ray projections for which an anatomical object had beenmeasured fromdiffer-ent directions. Allen MacLeod Cormack (–) and Sir Godfrey Hounsfield(–) are pioneers of medical computed tomography and in receivedthe Nobel Prize for Medicine for their epochal work during the s and s.Figure . shows a picture of the two Nobel Prize winners.

Table .. Summary of historical CT milestones

Year Milestone Röntgen discovers a new kind of radiation, which he named X-ray Röntgen receives the Nobel Prize for physics Bockwinkel employs the Lorentz’s solution in the reconstruction of three-

dimensional functions from two-dimensional area integrals Radon publishes his epochal work on the solution of the inverse problem of

reconstruction Ehrenfest extends the solution of Lorentz to n dimensions using the Fourier

transform Cramer and Wold solve the reconstruction problem in statistics in which the

probability distribution is obtained from a complete set of marginal probabilitydistributions

Eddington solves the reconstruction problem in the field of astrophysics to cal-culate the distribution of star velocities from the distribution of their measuredradial components

Bracewell applies Fourier techniques for the solution of the inverse problem inradio astronomy

TheUkrainian scientist Korenblyumdevelops anX-ray scanner and tries tomeas-ure thin slices through the patient with analogue reconstruction principles

Cormack contributes the firstmathematical implementations for tomographic re-construction in South Africa

Hounsfield shows proof of the principle with the first CT scanner based on a ra-dioactive source at the EMI research laboratories

Hounsfield andAmbrose publish the first clinical scanswith an EMIhead scanner Set-up of the first whole body scanner with a fan-beam system Hounsfield and Cormack receive the Nobel Prize for Medicine Demonstration of electron beam CT (EBCT) Kalender publishes the first clinical spiral-CT Demonstration of multi-slice CT (MSCT)

1.4 Some Examples

Cormack pointed out that previously the Dutch physicist H.A. Lorentz hadfound a solution to the three-dimensional problem in which the desired functionhad to be reconstructed from two-dimensional surface integrals (Cormack ).Lorentz himself did not publish the results and so, unfortunately, the context ofhis work is still unknown today. The result, however, is associated with Lorentz byH. Bockwinkel, whomentioned the work in a publication on light propagationin crystals.

A detailed, mathematical basis to the solution of the inverse problem in com-puted tomography was published by the Bohemian mathematician Johann Radonin (cf. Fig. ., left) (Radon ). Due to the complexity and depth of themathe-matical publication, however, the consequences of his ground-breaking results wererevealed very late in the mid-th century. Additionally, the paper was published inGerman, which hindered a wide distribution of the work. In �Chap. , an excerptof his original work is reprinted.

In Table ., some of the historical milestones and development steps of com-puted tomography are summarized.The list undoubtedly has to start with WilhelmConrad Röntgen (–), who received the Nobel Prize for physics in (cf.Fig. ., right). Before , a significant number of mathematical contributions inthe field of inverse problems were developed independently and are summarizedhere only retrospectively.

1.4Some Examples

Figures . to . show several examples of computer tomographic images that illus-trate different anatomical regions often used in clinical practice. Modern CT scan-ners yield images with an excellent soft tissue contrast. In Fig. ., the slices areannotated with the relevant scan protocol parameter. The most important parame-ters are the acceleration voltage (which determines the energy of the X-ray quanta),the tube current (which determines the intensity of the radiation), the slice thick-ness (which is the axial thickness of the X-ray fan beam), and the gantry tilt (whichis the angulation of the CT frame with respect to the axial axis). In spiral-CT, thepitch is an additional parameter that defines the table feed in units of slice thick-ness.

In clinical practice, besides choosing an appropriate set of scan parameters(cf. Figs. . and .), it is necessary to have a planning step for accurate anatom-ical scanning before CT slice sequence acquisition. In this planning step, the slicesmust be adapted to the anatomical situation, and furthermore, the dose for sensi-tive organs must be minimized. The planning is accomplished on the basis of anoverview scan that looks similar to simple projection radiography (cf. �Chap. ).Here, the exact position and orientation of the slice can be interactively defined.

1 Introduction

Fig. .. Examples of CT images. Modern CT scanners produce images with excellent softtissue contrast (courtesy of J. Ruhlmann)

Figures . and . illustrate that computed tomography is a three-dimensionalmodality. The geometrically precise slice stack can be constructed in a secondaryreconstruction step to yield a virtual three-dimensional volume. In Fig. ., five

1.4 Some Examples

Fig. .. A patient overview must be acquired for CT scan sequence planning. Dependingon the manufacturer the overview scan is called a topogram, a scout view, a scanogram ora pilot view.Thegeometric scan interval andgantry tilt are determined interactively (courtesyof J. Ruhlmann)

slices are illustrated as an example. Additionally, the patient’s skin and lung wassegmented with a simple threshold and visualized using a surface-rendering pro-cedure. For the same data set an alternative visualization is presented in Fig. ..Multi-planar reformatting (MPR) is used to show angulated sections through thethree-dimensional stack of slices. Typically, the principal sections (the sagittal, coro-nal, and axial slices) are presented to the radiologist. In Fig. ., the principal slicedirections are illustrated.

1 Introduction

Fig. .. Conventional CT produces two-dimensional slices. However, CT becomes a three-dimensional imaging modality if consecutive slices are arranged as axial stacks (courtesy ofJ. Ruhlmann)

Fig. ..Thearrangement of a set of axial CT slices to build up a three-dimensional volume iscalled secondary reconstruction.This data representation allows deeper diagnostic insights.Typically, segmented organs of interest are displayed using either surface rendering or anapproach in which the gray values are presented in an orthonormal reformatting consistingof the sagittal, coronal, and axial view (courtesy of J. Ruhlmann)

1.5 Structure of the Book

Fig. .. Primarily, with CT, an axial slice sequence is acquired and reconstructed. Usinginterpolation, coronal and sagittal slices can be calculated from the stack. This procedure iscalled multi-planar reformatting (MPR)

1.5Structure of the Book

This book gives a comprehensive overview of the main reconstruction methods incomputed tomography. The basis of the reconstruction is undoubtedly mathemat-ics. However, the beauty of computed tomography cannot be understood withouta basic knowledge of X-ray physics, signal processing concepts and measurementsystems. Therefore, the reader will find a number of references to these basic disci-plines as well as a brief introduction to many of the underlying principles.

With respect to the subtitle of this book, it is structured to cover the basics ofCT, from photon statistics to modern cone-beam systems. Without an elementaryknowledge of X-ray physics, a number of the described imaging effects and artifactscannot readily be understood. In �Chap. , X-ray generation, photon–matter inter-action, X-ray detection, and photon statistics are briefly summarized. In �Chap. ,a retrospective overview of the historical milestones on the road map of the tech-nical developments in computed tomography is given. Starting with tomosynthe-sis in the s and s, the different types or generations of CT are character-ized. The chapter concludes with motivation for the modern scanner concepts likeelectron-beam CT (EBCT), micro-CT, and especially helical cone-beam CT. Al-though remarkable advances in CT technology have been achieved, Fig. . showsthat the appearance of the gantry has undergone only a slight change throughoutthe years.

1 Introduction

Fig. ..Design of CT gantries in and (courtesy of Philips Medical Systems)

In �Chap. , the principles of signal processing are reviewed. This chapter fo-cuses on the necessary background of computed tomography and consequentlyuses signals of spatial variability. �Chapters and give a detailed overview oftwo-dimensional reconstruction mathematics. The most important algorithms arederived step by step. In �Chap. , the Fourier-based methods are collected. In�Chap. , the algebraic and statistical approaches are explained.

Fig. .a–c.Whole body scans can be performed with the latest generation of CT systems,including a multi-slice detector system. Even very small vessels of the feet can be preciselyvisualized (courtesy of Philips Medical Systems)

1.5 Structure of the Book

In �Chap. , the limitations of the practical implementation of the previouslydescribed methods are discussed. Specifically, the correspondence of the parallelpencil-beam and the fan-beam X-ray system are demonstrated.

In �Chap. , the three-dimensional methods of CT image or volume recon-struction are reviewed. It is shown that some of the ideas are consequent extensionsof the methods discussed in �Chap. .Themethods described in this chapter repre-sent the basis of a highly active field of research. A description of the existing mani-fold algorithmic variations in the field of helical cone-beam methods, for instance,is beyond the scope of this book. However, in Fig. . an example of the impressivequality of the three-dimensional reconstruction results of modern multi-slice CTscanners is given.

In �Chap. , an introduction to the methods of image quality evaluation isgiven. The chapter focuses on typical artifacts of computed tomography, wherebytwo-dimensional and three-dimensional artifacts are differentiated. Additionally,the important fourth power law is derived that describes the correspondence amongsignal-to-noise ratio, dose, and detector element size.

In �Chap. , some practical aspects of computed tomography are described.This includes CT planning, which uses the overview scan mentioned previously,the mapping of the physical attenuation values to the Hounsfield scale, and a listof exemplary application fields of CT in practice. Finally, �Chap. concludes thebook with a review of dose issues in clinical computed tomography.

2 Fundamentals of X-ray Physics

Contents

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 X-ray Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Photon–Matter Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4 Problems with Lambert–Beer’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.5 X-ray Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.6 X-ray Photon Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.1Introduction

For the discovery of a new radiation capable of high levels of penetration, WilhelmConrad Röntgen was awarded with the first Nobel Prize for physics in . In ,in experimentswith accelerated electrons, he had discovered radiationwith the abil-ity to penetrate optically opaque objects, which he named X-rays. In this chapter,the generation of X-rays, photon–matter interaction, X-ray detection, and statistic-al properties of X-ray quanta will be described. However, the scope of this chapteris limited to physical principles that are relevant to computed tomography (CT).A more comprehensive description can be found in many physics text books, forexample Demtröder (), and in overviews on radiological technology, for ex-ample Curry et al. (). One of the main reasons for the wide exploitation ofRöntgen’s radiation was the simple equipment required for X-ray generation anddetection. Nevertheless, the development of robust, high-power X-ray tubes thatare optimized for use in CT, is ongoing.

2.2X-ray Generation

X-ray radiation is of electromagnetic nature; it is a natural part of the electromag-netic spectrum, with a range that includes radio waves, radar and microwaves,infrared, visible and ultraviolet light to X- and γ-rays. In electron-impact X-ray

2 Fundamentals of X-ray Physics

sources, the radiation is generated by the deceleration of fast electrons enteringa solid metal anode, and consists of waves with a range of wavelengths roughlybetween −m and −m. Thus, the radiation energy depends on the electronvelocity, ν, which in turn depends on the acceleration voltage, Ua, between cathodeand anode so that with the simple conservation of energy

eUa = meν

(.)

the electron velocity can be determined.

2.2.1X-ray Cathode

In medical diagnostics acceleration voltages are chosen between kV and kV,for radiation therapy they lie between kV and kV, and for material testingthey can reach up to kV. Figure . shows a schematic drawing of an X-ray

Fig. .. Schematic drawing of an X-ray tube. Thermal electrons escape from a cathode fila-ment that is directly heated to approximately ,K. The electrons are accelerated in theelectric field between cathode and anode. X-ray radiation emerges from the deceleration ofthe fast electrons following their entry into the anode material

Charge of electrons: e = . ċ − C; mass of electrons: me = . ċ − kg. There is no clear definition of the X-ray wavelength interval. The range overlaps withultraviolet and γ-radiation.

Acceleration energy is measured in units called electron volts (eV). eV is the energy thatan electronwill gain if it is acceleratedby an electrical potential of one volt.The same unitis used to measure X-ray photon energy.

2.2 X-ray Generation

tube. Electrons are emitted from a filament, which is directly heated to approxi-mately ,K to overcome the binding energy of the electrons to the metal of thefilament .

The binding energy, Ev, is due to twomain effects (Bergmann and Schäfer ).The first effect is the formation of a dipole layer at the cathode surface. When a freeinner electron moves toward the surface of the metal, electrostatic forces preventthe electron from escaping. However, before reversing its direction, the electronovershoots the outer metal ion layer, and, as a result, an electron is missing insidethe metal for charge neutralization. The surplus of positive charge at the inner sideof the surface layer, together with the negative electron outside the metal, forms anelectric dipole layer. The electric field inside the dipole layer slows down electronstrying to leave the metal.This effect results in the partWDipole of the work function.

The other part originates fromwhat is called a mirror-image force. Due to elec-trostatic influence, an electron above a metal surface causes a charge displacementinside the metal. The resulting electric field between the electron and the metal sur-face looks like the field between a charge, −e, above and a virtual mirror charge, +e,below the metal boundary at the same distance x from the boundary. To bring anelectron from distance d above the surface to infinity, the work

Wmirror = e

πε

�

∫d

dx(x) = e

πεd(.)

Fig. ..The electron beam is controlled by a cylindrically shaped electrode, containing thecathode with opposite potential.This electron optics is called aWehnelt cylinder, or is some-times also called a focusing cup. In this way, the electrons are steered onto a small focalpoint on the anode. Shown in a is a dual-filament and in b a modern mono-filament; bothare designed to produce focal spot sizes of .mm and .mm (courtesy of Philips MedicalSystems)

Filaments are usually made of thoriated tungsten with a melting point at ,�C.

2 Fundamentals of X-ray Physics

must be applied. Since the metal–vacuum interface is not an ideal mirror surface, itis sensible to start the integration at a distance of approximately one atom diameter(d � −m), leading to a work of Wmirror � .eV.

Due to their thermal energy, electrons are boiled off from the filament. Thisprocess is called thermionic emission. The temperature of the metal must be highenough to increase the kinetic energy, Ekin, of the electrons such that Ekin �Ev = Wdipole + Wmirror. The emission current density, je, is essentially a func-

Fig. .. Simulation of electron trajectories emitted from the filament and accelerated ontothe anode. The potential at the Wehnelt cylinder controls the electron focus on the anode.Below: Shape and size of a large and a small X-ray focus (courtesy of Philips Medical Sys-tems)

2.2 X-ray Generation

tion of the temperature and can be described by the Richardson–Dushman equa-tion

je = CRDT e−φkT , (.)

where CRD is the Richardson–Dushman constant

CRD = πmekeh , (.)

k is the Boltzmann constant and φ is the work function defined as the differencebetween Ev and the Fermi energy edge. An electron cloud forms around the fila-ment and these electrons are subsequently accelerated toward the anode. When theelectrons reach the surface of the anode they will be stopped abruptly.

To produce a small electron focus on the anode, the trajectories of the acceler-ated electronsmust be controlled by electron optics.The focusing device can be seenin Fig. .. Basically, it is a cup-shaped electrode that forms the electric field nearthe filaments such that the electron current is directed to a small spot. In Fig. .it can be seen how the potential of the Wehnelt optics (frequently named Wehneltcylinder) influences the electron trajectories. In this way, the cylinder can easily becontrolled to produce a large or small X-ray focus.The effect that this has on imag-ing quality will be discussed in a later section.

2.2.2Electron–Matter Interaction

With the entry of accelerated electrons into the anode, sometimes also called theanticathode, several processes take place close to the anode surface. Generally, theelectrons are diffracted and slowed down by the Coulomb fields of the atoms in theanode material. The deceleration results from the interaction with the orbital elec-trons and the atomic nucleus. As known from classical electrodynamics, accelera-tion and deceleration of charged particles creates an electric dipole and electromag-netic waves are radiated. Usually, several photons emerge throughout the completedeceleration process of one single electron. Figure .a illustrates two successive de-celeration steps. It can happen, however, that the entire energy, eUa, of an electronis transformed into a single photon. This limit defines the maximum energy of theX-ray radiation, which can be determined by

eUa = hνmax = Emax . (.)

The limit Emax corresponds to the minimum wavelength

λmin = hceUa

= .nmUa�kV , (.)

For ideal metals CRD � A cm− K−. However, in practice CRD is material-dependent. Boltzmann constant: k = . ċ − J K− . .eV for tungsten.

2 Fundamentals of X-ray Physics

Fig. ..X-ray spectrumof a tungsten anode at accelerationvoltages in the range ofUa = –kV. The anode angle is � and mm Al filtering has been applied. The intensity versuswavelength plot shows the characteristic line spectrum as well as the continuous bremsstrah-lung (courtesy of B. David, Philips Research Labs). The minimumwavelength is determinedby the total energy, eUa, of the electron reaching the anode. Process illustrations: a brems-strahlung, b characteristic emission, c Auger process and d direct electron-nucleus collision

where h is Planck’s constant and c is the speed of light. While the accelerationvoltage determines the energy interval of the X-ray spectrum, the intensity of thegenerated X-ray spectrum or the number of X-ray quanta, is solely controlled bythe anode current. Planck’s constant: h = . ċ − Js; speed of light in vacuum: c = . ċ m�s.